Question: All of the 3rd grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.50$ each for students, and the group paid $$49.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$22.00$ each for teachers and $$11.00$ each for students, and the group paid $$154.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4.5y = 49}$ ${22x+11y = 154}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-18y = -196}$ ${22x+11y = 154}$ Add the top and bottom equations together. $ -7y = -42 $ $ y = \dfrac{-42}{-7}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5.5x+4.5y = 49}$ to find $x$ ${5.5x + 4.5}{(6)}{= 49}$ $5.5x+27 = 49$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {22x+11y = 154}$ and get the same answer for $x$ ${22x + 11}{(6)}{= 154}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.